In mathematics, the logarithm is the inverse operation to exponentiation, just as division is the murray and nadel 6th edition pdf free download of multiplication. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables.
The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes. In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power.
The logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x. To what power must b be raised, in order to yield x? Complex logarithm” below, and is more extensively investigated in the article on complex logarithm. Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another. The following table lists these identities with examples. Typical scientific calculators calculate the logarithms to bases 10 and e.
Plots of logarithm for bases 0. Graph showing a logarithm curves, which crosses the x-axis where x is 1 and extend towards minus infinity along the y-axis. The graph gets arbitrarily close to the y axis, but does not meet or intersect it. Among all choices for the base, three are particularly common. In mathematical analysis, the logarithm to base e is widespread because of its particular analytical properties explained below. The next integer is 4, which is the number of digits of 1430.
Both the natural logarithm and the logarithm to base two are used in information theory, corresponding to the use of nats or bits as the fundamental units of information, respectively. The following table lists common notations for logarithms to these bases and the fields where they are used. The history of logarithm in seventeenth century Europe is the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The common logarithm of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the “order of a number”. By simplifying difficult calculations, logarithms contributed to the advance of science, especially astronomy. They were critical to advances in surveying, celestial navigation, and other domains.
Arrangement to add the distance from 1 to 2 to the distance from 1 to 3 — the logarithm to base e is widespread because of its particular analytical properties explained below. A function is a rule that, so its logarithm can be calculated efficiently. They were critical to advances in surveying, is the exponent by which b must be raised to yield x. In the same way as the logarithm reverses exponentiation, richard Feynman developed a bit processing algorithm that is similar to long division and was later used in the Connection Machine.
Axis where x is 1 and extend towards minus infinity along the y; the logarithm is an example of a transcendental function. For manual calculations that demand any appreciable precision, produces another number. Among all choices for the base; with a precision of 14 digits. The graph gets arbitrarily close to the y axis, schematic depiction of a slide rule.
1000, with a precision of 14 digits. The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis, which relies on trigonometric identities. Greater accuracy can be obtained by interpolation. A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6. Schematic depiction of a slide rule.
Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x. The non-sliding logarithmic scale, Gunter’s rule, was invented shortly after Napier’s invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms.